First welcome to the forums. Its too bad it has to be for an issue like this. Second, your contacts should all be on Google so you won't need to back them up. Third, all user data for apps and stuff are stored on the SD card which can be swapped out when you get the new phone. Or just copy all your data from the card to the computer and then back to the new one when you get it. (and yes you need to use a micro sd card adapter since your phone is not working)
I really hope this at least will help you save your data. Really sucks to hear about the breakage.
Oh and purchased apps on google will still be there when you re-log into the new phone in the downloads page of the market.